An Introduction to Fourier and Wavelet Analysis: Part I

1. An Introduction to Fourier and Wavelet Analysis:Part I Norman C. Corbett Sunday, September 28, 2014

2. Motivation • Search for patterns, periodicity or recurrent features in nature • An important element in the (stable) behaviour of many physical systems • Examine the long profile of rivers • Contribute to the debate about the periodicity of step-pool sequences

3. Signals • Variety of definitions • Real valued functions of one or more independent variables • Serve as mathematical models for quantities that vary in time and/or space • Denote a generic signal by s(t)

4. Signals • Some examples: • The voltage v(t) in an electrical circuit as a function of time t • A black and white image I(x,y) • What about a colour image? • The density r(s) of tissue along a curvilinear path through a human brain • The long profile of a river E(d)

5. A Diatribe of Periodicity • No real world signal is strictly periodic! • Noise, friction, etc. crash the party. • Check periodicity by constructing a phase plane plot: • It’s easy to make an a-periodic signal s1 s2

6. Sinusoids • The search for periodicity is the search for the sinusoidal components of a signal • General sinusoid • AmplitudeA (units of observed quantity) • Frequencyf (Hz: cycles/s) or wave numbern (1/m) • Angular frequency w =2p f (radians/s) • PeriodT (s) or wavelengthl (m) • Phase anglef (radians)

7. An Example • What sinusoidal components do you see in the graph of? • That was easy. What about this one? • Picking out the periodic components of a general signal is very difficult All real world signals contain noise! s1 s2

8. Fourier to the Rescue! • The continuous Fourier transform (CFT) where • “Compare” s(t) to a family of complex exponentials indexed by frequency f • CFT of noisy sinusoid Fs2

9. Another Example • Define s(t) by rectangular pulse • The CFT is

10. Complex Numbers • The CFT s(t) is complex number. • A complex number is a point in the complex plane • Real parta • Imaginary partb • Magnitude • Argument

11. Spectrum • Since the CFT is complex, we compute • Amplitude spectrum • Spectral density (energy/power) • Sometimes look at phase spectrumf ( f ) • Amplitude spectrum of the rectangular pulse AS

12. Inverse Fourier Transform • The original signal can be recovered via • Implies that s(t) can be expressed as a “weighted sum” of complex exponentials • The magnitude of the weight is the amplitude spectrum

13. Computation • In practice we work with samples of s(t) on a finite interval [0,T] • Ts=T/N is the sampling period • Discrete Fourier transform (DFT)

14. Computation • Under suitable conditions • We can use the DFT to estimate the CFT at the discrete frequencies: est

15. Some Real World Signals • Look at the spectra of river profiles in terms of the periods (wavelength) • A smackrel of weather data: • Winnipeg monthly mean temperatures for the years Zim Bur Wpg

16. A Problem with Fourier • Real world signals are often nonstationary • The spectra of such signals are very complex • Detrending ensures first orderstationarity • The mean of the residuals is zero • Second and higher order moments may still evolve: • Variance, (auto)correlation, etc.

17. Yet Another Example • The two signals • Have similar amplitude spectra • Remove the noise and look at the underlying signals • The frequency of the second signal changes • Can’t see this by looking at spectra • Wavelets to the rescue! (TBC) s2 s3 Fs2 Fs3 cs2 cs3

18. Questions?